Theorem coep 32989

Description: Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)

Hypotheses

Ref	Expression
coep.1	⊢ 𝐴 ∈ V
coep.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
coep	⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem coep

Step	Hyp	Ref	Expression
1		coep.2	. . . . 5 ⊢ 𝐵 ∈ V
2	1	epeli 5470	. . . 4 ⊢ (𝑥 E 𝐵 ↔ 𝑥 ∈ 𝐵)
3	2	anbi1ci 627	. . 3 ⊢ ((𝐴𝑅𝑥 ∧ 𝑥 E 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥))
4	3	exbii 1848	. 2 ⊢ (∃𝑥(𝐴𝑅𝑥 ∧ 𝑥 E 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥))
5		coep.1	. . 3 ⊢ 𝐴 ∈ V
6	5, 1	brco 5743	. 2 ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝑥 E 𝐵))
7		df-rex 3146	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝐴𝑅𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝐴𝑅𝑥))
8	4, 6, 7	3bitr4i 305	1 ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥)

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1780   ∈ wcel 2114  ∃wrex 3141  Vcvv 3496   class class class wbr 5068   E cep 5466   ∘ ccom 5561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-eprel 5467  df-co 5566

This theorem is referenced by:  dffr5  32991  brbigcup  33361  elfuns  33378  brimage  33389